Journal of Prime Research in Mathematics Vol. 4(2008), 181-194 ALMOST PERIODIC FUNCTIONS DEFINED ON R n WITH VALUES IN LOCALLY CONVEX SPACES MUHAMMAD AMER LATIF ∗ , MUHAMMAD IQBAL BHATTI ∗∗ Abstract. In this paper we develop the theory of almost periodic func- tions defined on R n with values in locally convex spaces and Fr´ echet spaces. Key words : Almost periodic functions, locally convex spaces, Fr´ echet spaces, relatively dense set. AMS SUBJECT: Primary 05C38, 15A15; Secondary 05A15, 15A18. 1. Introduction The theory of almost periodic functions was started between 1923 and 1926 by Harald Bohr (1887-1951) (13). The theory has attracted many mathemati- cians for decades. Some generalizations of the concept have been introduced successfully by V.V. Stepanov and A.S Besicovitch (4). Almost periodic func- tions with values in Banach spaces was first studied by S. Bochner (16) and developed by many mathematicians like C. Corduneanu (8), S. Zaidman (17) , J.A. Goldstein, L. Amerio, G. Prouse (15), and A.M. Fink (2) etc. The the- ory of almost periodic functions defined on R n with values in a locally convex space or in a Fr´ echet space was first studied by G.M. N Gu´ er´ ekata e.g see (9, 10, 11, 12). In the monograph of S. Zaidman (17) the theory of almost peri- odic functions defined on R n with values in Banach spaces is elaborated but the theory of almost periodic functions defined on R n with values in Locally convex spaces or Fr´ echet spaces was not developed yet. In this paper we define almost periodicity of functions which are defined on R n and have values in a locally convex space or in a Fr´ echet space and give some results. ∗ Aeronautical Training Center, General Authority of Civil Aviation P.O Box, 15441, Jeddah, 21444, Saudi Arabia . E-mail: m amer latif@hotmail.com, ∗∗ University of Engineering and Technology Lahore, Pakistan. E-mail: iqbalb- hatti@uet.edu.pk. 181